Some Remarks On The Convergence Of Picard

Some examples are given to illustarte that the characterization in [8] for the convergence of Picard iteration to a fixed point for a continuous mapping, is false. Throughout this paper, let T be a mapping from a nonempty subset D of a metric space X into X and let F (T) be a fixed point set of T. A mapping T: D → X is called nonexpansive if for each x, y ∈ D, d(T (x), T (y)) ≤ d(x, y). T is said to be quasi-nonexpansive if for each x ∈ D and for every p ∈ F (T), d(T (x), p) ≤ d(x, p). T is conditionally quasi-nonexpansive if it is quasi-nonexpansive whenever F (T) 6 = ∅. Since convergence theorems of iterations to a fixed point for nonexpansive mapping are discussed in [1-2], many results on the convergence of some iterations to fixed points for nonexpansive, quasi-nonexpansive and generalized types of quasi-nonexpansive map-pings in metric and Banach spaces have appeared (for example, [3-8]). Following Ghosh and Debnath [4], if D is a convex subset of a normed space X and T: D → D, Ishikawa introduced the following iteration x0 ∈ D, xn = Tnλ,µ(x0), Tλ,µ = (1 − λ)I + λT [(1 − µ)I + µT], for each n ∈ N (the set of all positive integers), where λ ∈ (0, 1) and µ ∈ [0, 1). When µ = 0, it yields that Tλ,µ = Tλ and the iteration becomes x0 ∈ D, xn = Tnλ (x0), Tλ = (1 − λ)I + λT. This iteration is called Mann iteration. If Tµ = (1 − µ)I + µT, Tλ,µ may be written in the form Tλ,µ = (1 − λ)I + λTTµ. Recall a mapping T is asymptotically regular at x0 ∈ D if lim n→ ∞ d(T n(x0), Tn+1(x0)) = 0